# Just Hammond

This article features just intervals created by the mechanical tonegenerator of the classical Hammond B-3 Organ model.

## Design of the Hammond B-3’s Tonegenerator

Since 1935 the Hammond Organ Company’s goal was to market electromechanical organs^{[1]} with 12-tone equally tempered (12edo) tuning. The mechanical tonegenerator of the Hammond B-3 Organ is based on a set of twelve *different pairings* of gearwheels that make (12*4) *driven* shafts turn. The corresponding *driving* gearwheels are mounted on a common shaft and turn all at the same rotational speed *n _{1}*. Certain gears reduce, others increase rotational speed.

^{[2]}

For every chromatic pitch class four driven shafts are installed. Pure octaves are generated by dedicated *tonewheels* (with 2, 4, 8, 16, 32, 64 or 128 high and low points on their edges) that rotate with the driven shafts. Each high point on a tone wheel is called a *tooth*. When the gears are in motion, magnetic pickups react to the tonewheels’ passing teeth and generate an electric signal that can be amplified and transmitted to a loudspeaker.

For each pair of gearwheels the ratio of rotational speed *n _{2}*/

*n*is determined by the inverse ratio of the gearwheels’ integer teeth numbers Z

_{1}_{1}and Z

_{2}:

[math]\frac{Z_1}{Z_2}=\frac{n_2}{n_1}[/math]

To calculate the rotational speed *n _{2}* of the driven shafts we write

[math]n_2=\frac{Z_1}{Z_2}\cdot n_1[/math]

__Table 1__: Pairings of Gearwheels^{[3]} / Ratios and Intervals

Pitch
Class |
HAMMOND |
Gear Ratio |
Conversion |
Deviation |
Intonation | |||
---|---|---|---|---|---|---|---|---|

(A) | (B) | (C) | (D) | |||||

driving
Z |
driven
Z |
Fraction |
Ratio
(C)/(D) |
[cents] |
[cents] |
[cents] | ||

C | 85 | 104 | 85 | 104 | 0.8173077 | -349.26 | -49.26 | -0.576 |

C# | 71 | 82 | 71 | 82 | 0.8658537 | -249.37 | -49.37 | -0.684 |

D | 67 | 73 | 67 | 73 | 0.9178082 | -148.48 | -48.48 | 0.200 |

D# | 105 | 108 | 35 | 36 | 0.9722222 | -48.77 | -48.77 | -0.088 |

E | 103 | 100 | 103 | 100 | 1.0300000 | 51.17 | -48.83 | -0.145 |

F | 84 | 77 | 12 | 11 | 1.0909091 | 150.64 | -49.36 | -0.681 |

F# | 74 | 64 | 37 | 32 | 1.1562500 | 251.34 | -48.66 | 0.026 |

G | 98 | 80 | 49 | 40 | 1.2250000 | 351.34 | -48.66 | 0.020 |

G# | 96 | 74 | 48 | 37 | 1.2972973 | 450.61 | -49.39 | -0.707 |

A | 88 | 64 | 11 | 8 | 1.3750000 | 551.32 | -48.68 | 0.000 |

A# | 67 | 46 | 67 | 46 | 1.4565217 | 651.03 | -48.97 | -0.285 |

B | 108 | 70 | 54 | 35 | 1.5428571 | 750.73 | -49.27 | -0.593 |

(Purple colored cells contain prime numbers)

## Just Intervals

When we associate *“ratios of the gearwheels’ integer teeth numbers”* with *“frequency ratios between partials”* we realize an intrinsic *just interval* determined by integer teeth numbers within such mechanical gear - even without turning the shafts! Although the Hammond Organ pretends to generate a 12edo scale, the instrument in fact creates a *high prime limit* just scale.

## Tuning

The whole set of frequency *ratios* is fixed by the design of the gear mechanism. The driving shaft’s (A) rotational speed *n _{1}* determines the instrument’s (master-) tuning. Rotating at exactly 1200 rpm (20 rev./sec), the pitch of note A equals precisely 27.500 Hz or one of its doublings. Therefore the instrument aligns note A with a concert pitch of 440.0 Hz.

[math]f_\text{A}=20.0/\text{s}\cdot\frac{88}{64}\cdot(2^4)=440.0/\text{s} = 440.0 \text{ Hz}[/math]

## Mapping Hammond’s Rational Intervals to the Harmonic Series

To find out, where the rational intervals played on a Hammond Organ occur in the harmonic series we

- cancel the fractions of gear-ratios specified by Hammond and
- calculate the
*least common multiple (LCM)*of the denominators of "*intervals of interest"*by prime factorization - With this specific LCM we recalculate the numerators of the intervals. The resulting numerators correspond to the partial numbers we are looking for.

Before we proceed, we have to agree on a numbering scheme for octaves in the harmonic series.

## Numbering Octaves

We apply the scheme from the article *First Five Octaves of the Harmonic Series* and number the octaves as follows:

- Integer octave numbering starts with
**#1**for the range between the 1st and < 2nd partial

- The
**2nd**octave starts at partial #2 (= 2^{1}) and covers partials 2 and 3 - The
**3rd**octave starts at partial #4 (= 2^{2}) and covers partials 4, 5, 6 and 7 - The
**4th**octave starts at partial #8 (= 2^{3}) and covers partials 8, 9, 10, 11, 12, 13, 14 and 15. - ...

This numbering scheme is consistent with the scheme used by Bill Sethares^{[4]} : “*In general, the n ^{th} octave contains 2^{n-1} pitches*.”

## Mapping Hammond’s Rational Intervals (cont.): Examples

The following examples illustrate how to map intervals or chords to the harmonic series.

#### Example 1: Mapping a single interval

In this example we map the combination of a Hammond Organ’s note E and a higher note A to the harmonic series.

__Table 2__: Mapping the fourth E-A

Pitch |
HAMMOND |
Prime |
Ascending |
Partial Found | |||||
---|---|---|---|---|---|---|---|---|---|

(C) | (D) |
...of Column (D) |
Recalculated |
Counted | |||||

Fraction (C)/(D) | |||||||||

E | 103 | 100 | 2 | 2 | 5 | 5 | 206 | 8.7 | |

A | 11 | 8 | 2 | 2 | 2 | 275 | 9.6 | ||

Multiply --------> |
2 |
2 |
2 |
5 |
5 |
200 |
8.6 |

The resulting interval appears between partial # 206 and partial # 275. Thus the frequency ratio is (275:206), which equals 500.14 cents.

#### Example 2: Mapping a chord

Adding an upper fifth (note B), the second example illustrates how to map the resulting sus4-chord E-A-B to the harmonic series.

__Table 3__: sus4-chord E-A-B

Pitch |
HAMMOND |
Prime |
Ascending |
Partial found | ||||||
---|---|---|---|---|---|---|---|---|---|---|

(C) | (D) |
...of Column (D) |
Recalculated |
Counted | ||||||

Fraction (C)/(D) | ||||||||||

E | 103 | 100 | 2 | 2 | 5 | 5 | 1442 | 11.5 | ||

A | 11 | 8 | 2 | 2 | 2 | 1925 | 11.9 | |||

B | 54 | 35 | 5 | 7 | 2160 | 12.1 | ||||

Multiply --------> |
2 |
2 |
2 |
5 |
5 |
7 |
1400 |
11.4 |

The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11^{th} and the 12^{th} octave.

#### Example 3: Mapping all of the tonegenerator's pitchclasses

The full set of the Hammond Organ’s intervals resides surprisingly far up in the Harmonic Series:

- The
**44th****octave starts**at partial #(2^{43}), just below the set of partials determined by the Hammond Organ’s tonegenerator - The
**45th****octave**starts**right within the derived set of partials**and starts at partial #(2^{44})

__Table 4__: The full set of intervals' position in the Harmonic Series

Pitch |
HAMMOND |
Prime |
Ascending |
Partial Found | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(C) | (D) |
... of Column (D) |
Recalculated |
Counted from | ||||||||||||||||

Fraction (C)/(D) | ||||||||||||||||||||

C | 85 | 104 | 2 | 2 | 2 | 13 | 15,003,356,791,500 | 44.8 | ||||||||||||

C# | 71 | 82 | 2 | 41 | 15,894,517,438,800 | 44.9 | ||||||||||||||

D | 67 | 73 | 73 | 16,848,249,818,400 | 44.9 | |||||||||||||||

D# | 35 | 36 | 2 | 2 | 3 | 3 | 17,847,130,301,000 | 45.0 | ||||||||||||

E | 103 | 100 | 2 | 2 | 5 | 5 | 18,907,759,758,888 | 45.1 | ||||||||||||

F | 12 | 11 | 11 | 20,025,870,883,200 | 45.2 | |||||||||||||||

F# | 37 | 32 | 2 | 2 | 2 | 2 | 2 | 21,225,337,107,975 | 45.3 | |||||||||||

G | 49 | 40 | 2 | 2 | 2 | 5 | 22,487,384,179,260 | 45.4 | ||||||||||||

G# | 48 | 37 | 37 | 23,814,549,158,400 | 45.4 | |||||||||||||||

A | 11 | 8 | 2 | 2 | 2 | 25,240,941,425,700 | 45.5 | |||||||||||||

A# | 67 | 46 | 2 | 23 | 26,737,439,929,200 | 45.6 | ||||||||||||||

B | 54 | 35 | 5 | 7 | 28,322,303,106,240 | 45.7 | ||||||||||||||

Multiply --------> |
2 |
2 |
2 |
2 |
2 |
3 |
3 |
5 |
5 |
7 |
11 |
13 |
23 |
37 |
41 |
73 |
18,357,048,309,600 |
45.1 |

## Discussion

No doubt - the evidence that a cluster of 12 simultaneously ringing semitones from a Hammond Organ is allocated around the 45^{th} octave of the harmonic series is of limited practical value. The intervals' *far-up placement* is mainly caused by Laurens Hammond’s implementation of various prime numbers (11, 13, 23, 37, 41, 73) in different gearwheel pairings.

- Respective high-order partials are very densely spaced (in the range of
*pico-cents)*and intervals between successive partials up there are too narrow for musical applications by far

- Due to its construction the tonegenerator selects only twelve from 17.6 trillion varieties in the 45
^{th}octave where…- the partial number associated with the LCM, which is located exactly 8/11 below pitch class A, is not addressed because there is no gear with transmission ratio 1.000
- no pure octave above a virtual root (1/1; partial# (2
^{44})) is playable, which would ring -624.997 cents way down from pitchclass A

## General Applicability

The method of prime factorization to find the LCM can be applied to arbitrary **intervals, chords or scales built from rational intervals** to identify their position in the harmonic series. Simply replace the gear-ratios by just intervals of interest.

## References

- ↑ Webressource https://en.wikipedia.org/wiki/Hammond_organ (retrieved December 2019)
- ↑ Detailed photos of a similar M-1 tonegenerator are provided by https://modularsynthesis.com/hammond/m3/m3.htm (retrieved December 2019)
- ↑ Gearing details were taken from http://www.goodeveca.net/RotorOrgan/ToneWheelSpec.html (retrieved Dec 29, 2019)
The German Wikipedia provides the same technical information (in German): https://de.wikipedia.org/wiki/Hammondorgel#Tonerzeugung (retrieved Dec 29, 2019)
The
*HammondWiki*publishes a second, alternative set of gear ratios with slightly deviating pitch class “E”. Certain other pitch classes are shifted by pure octaves. http://www.dairiki.org/HammondWiki/GearRatio (retrieved Dec 29, 2019) - ↑ Sethares, William A.
*Tuning Timbre Spectrum Scale.*London: Springer Verlag , 1999.

## See also…

- Dismantling the tonegenarator of a scrapped H-Series Hammond Organ [8:47 min]

https://www.youtube.com/watch?v=7Qqmr6IiFLE

- An artist’s perception: Tony Monaco demonstrates how to apply the tonegenerator’s features of a Hammond Organ [31:10 min]

https://www.youtube.com/watch?v=5CG81_Y8SvY

- @ 4:48 min:
*“…these sounds are in there”* - @ 5:40 min:
*“16 foot, biggest pipes, the deepest sounds – they come from the foot”*